C-Spline Basis for Polynomial SplinesSource:
Generates the convex regression spline (called C-spline) basis matrix by integrating I-spline basis for a polynomial spline or the corresponding derivatives.
cSpline( x, df = NULL, knots = NULL, degree = 3L, intercept = TRUE, Boundary.knots = NULL, derivs = 0L, scale = TRUE, ... )
The predictor variable. Missing values are allowed and will be returned as they are.
Degree of freedom that equals to the column number of the returned matrix. One can specify
knots, then the function chooses
df - degree - as.integer(intercept)internal knots at suitable quantiles of
xignoring missing values and those
xoutside of the boundary. If internal knots are specified via
knots, the specified
dfwill be ignored.
The internal breakpoints that define the splines. The default is
NULL, which results in a basis for ordinary polynomial regression. Typical values are the mean or median for one knot, quantiles for more knots.
The degree of C-spline defined to be the degree of the associated M-spline instead of actual polynomial degree. For example, C-spline basis of degree 2 is defined as the scaled double integral of associated M-spline basis of degree 2.
TRUEby default, all of the spline basis functions are returned. Notice that when using C-Spline for shape-restricted regression,
intercept = TRUEshould be set even when an intercept term is considered additional to the spline basis in the model.
Boundary points at which to anchor the splines. By default, they are the range of
NA. If both
Boundary.knotsare supplied, the basis parameters do not depend on
x. Data can extend beyond
A nonnegative integer specifying the order of derivatives of C-splines. The default value is
0Lfor C-spline basis functions.
A logical value indicating if scaling C-splines is required. If
TRUEby default, each C-spline basis is scaled to have unit height at right boundary knot. The corresponding I-spline and M-spline produced by
derivmethods will be scaled to the same extent.
Optional arguments that are not used.
A numeric matrix of
length(x) rows and
df columns if
df is specified or
length(knots) + degree +
as.integer(intercept) columns if
knots are specified instead.
Attributes that correspond to the arguments specified are returned
mainly for other functions in this package.
It is an implementation of the closed-form C-spline basis derived from the recursion formula of I-splines and M-splines.
Meyer, M. C. (2008). Inference using shape-restricted regression splines. The Annals of Applied Statistics, 2(3), 1013--1033.
library(splines2) x <- seq.int(0, 1, 0.01) knots <- c(0.3, 0.5, 0.6) ### when 'scale = TRUE' (by default) csMat <- cSpline(x, knots = knots, degree = 2) op <- par(mar = c(2.5, 2.5, 0.2, 0.1), mgp = c(1.5, 0.5, 0)) matplot(x, csMat, type = "l", ylab = "C-spline basis") abline(v = knots, lty = 2, col = "gray") isMat <- deriv(csMat) msMat <- deriv(csMat, derivs = 2) matplot(x, isMat, type = "l", ylab = "scaled I-spline basis") matplot(x, msMat, type = "l", ylab = "scaled M-spline basis") ## reset to previous plotting settings par(op) ### when 'scale = FALSE' csMat <- cSpline(x, knots = knots, degree = 2, scale = FALSE) ## the corresponding I-splines and M-splines (with same arguments) isMat <- iSpline(x, knots = knots, degree = 2) msMat <- mSpline(x, knots = knots, degree = 2, intercept = TRUE) ## or using deriv methods (more efficient) isMat1 <- deriv(csMat) msMat1 <- deriv(csMat, derivs = 2) ## equivalent stopifnot(all.equal(isMat, isMat1, check.attributes = FALSE)) stopifnot(all.equal(msMat, msMat1, check.attributes = FALSE))